Tree-chromatic number is not equal to path-chromatic number

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $\chi(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are different. This settles a question of Seymour. Our results also imply that the path-chromatic numbers of the Mycielski graphs are unbounded.Comment: 11 pages, 0 figure

Tree-chromatic number

Abstract Let us say a graph G has "tree-chromatic number" at most k if it admits a tree-decomposition (T, (X t : t ∈ V (T ))) such that G[X t ] has chromatic number at most k for each t ∈ V (T ). This seems to be a new concept, and this paper is a collection of observations on the topic. In particular we show that there are graphs with tree-chromatic number two and with arbitrarily large chromatic number; and for all ℓ ≥ 4, every graph with no triangle and with no induced cycle of length more than ℓ has tree-chromatic number at most ℓ − 2